Aristotle's axiom

Aristotle's axiom is an axiom in the foundations of geometry, proposed by Aristotle in On the Heavens that states: If

is an acute angle and AB is any segment, then there exists a point P on the ray

, such that PQ is perpendicular to OX and PQ > AB.

Aristotle's axiom is a consequence of the Archimedean property,[1] and the conjunction of Aristotle's axiom and the Lotschnittaxiom, which states that "Perpendiculars raised on each side of a right angle intersect", is equivalent to the Parallel Postulate.

[2] Without the parallel postulate, Aristotle's axiom is equivalent to each of the following two incidence-geometric statements:[3] [4]

Aristotle's axiom asserts that a line PQ exists which is parallel to AB but greater in length. Note that: 1) the line AB does not need to intersect OY or OX; 2) P and Q do not need to lie on the lines OY and OX, but their rays (i.e. the infinite continuation of these lines).